Variance Estimation

The survey analysis procedures provide a choice of variance estimation methods for complex survey designs. In addition to
the Taylor series linearization method, the procedures offer two replication-based (resampling) methods—balanced repeated
replication (BRR) and the delete-1 jackknife.
These variance estimation methods usually give similar, satisfactory results (Lohr, 2010; Särndal, Swensson, and Wretman, 1992; Wolter, 2007). The choice of a variance estimation method can depend on the sample design used, the sample design information available,
the parameters to be estimated, and computational issues. See Lohr (2010) for more details.

The Taylor series linearization method is appropriate for all designs where the first-stage sample is selected with replacement,
or where the first-stage sampling fraction is small, as it often is in practice. The Taylor series method obtains a linear
approximation for the estimator and then uses the variance estimate for this approximation to estimate the variance of the
estimate itself (Fuller, 1975; Woodruff, 1971). When there are clusters (PSUs) in the sample design, the procedures estimate the variance from the variation among the
PSUs. When the design is stratified, the procedures pool stratum variance estimates to compute the overall variance estimate.

For a multistage sample design, the Taylor series method uses only the first stage of the sample design. Therefore, the required
input includes only first-stage cluster (PSU) and first-stage stratum identification. You do not need to input design information
about any additional stages of sampling.

Replication methods for variance estimation draw multiple replicates (or subsamples) from the full sample by following a specific
resampling scheme. Commonly used resampling schemes include balanced repeated replication (BRR) and the jackknife. The parameter of interest is estimated from each replicate, and the variability among the replicate estimates is used to
estimate the overall variance of the parameter estimate.

The BRR variance estimation method requires a stratified sample design with two PSUs in each stratum. Each replicate is obtained
by deleting one PSU per stratum according to the corresponding Hadamard matrix and adjusting the original weights for the
remaining PSUs. The adjusted weights are called replicate weights. The survey procedures also provide Fay’s method, which is a modification of the BRR method.

The jackknife method deletes one PSU at a time from the full sample to create replicates, and modifies the original weights
to obtain replicate weights. The total number of replicates equals the number of PSUs. If the sample design is stratified,
each stratum must contain at least two PSUs, and the jackknife is applied separately within each stratum.

Instead of having the survey procedures generate replicate weights for the analysis, you can directly input your own replicate
weights. This can be useful if you need to do multiple analyses with the same set of replicate weights, or if you have access
to replicate weights without complete design information.